Check that the ratio $k{e}^2$ /G $m_e{m}_p$ is dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?

The given ratio is $\frac{k{e}^2}{G{m}_c{m}_p}$

Where,

$G=$ Gravitational constant

Its unit is $N{m}^{2}k{g}^{-2}$.

$m_e$ and $m_p=$ Masses of electron and proton.

Their unit is kg.

$e=$ Electric charge.

Its unit is C.

$k=A$ constant

   $=\frac{1}{4\pi {\in }_0}$

${\in }_0=$ Permittivity of free space

Its unit is $N{m}^2{C}^{-2}$.

 Unit of the given ratio $\frac{k{e}^2}{G{m}_c{m}_p}=\frac{[N{m}^2{C}^2][C^{-2}]}{[N{m}^{2}k{g}^{-2}][kg][kg]}$

                                                                        $=M^0{L}^0{T}^0$

Hence, the given ratio is dimensionless.

$e=1.6\times 10^{-19}C$

$G=6.67\times 10^{-11}N{m}^{2}k{g}^{-2}$

$m_e=9.1\times 106-31kg$

$m_p=1.66\times 10^{-27}kg$

$\frac{k{e}^2}{G{m}_c{m}_p}=\frac{9\times 10^9\times (1.6\times 10^{-19}{)}^2}{6.67\times 10^{-11}\times 9.1\times 10^{-3}\times 1.67\times 10^{-22}}\approx 2.3\times 10^{39}$

This is the ratio of electric force to the gravitational force between a proton and an electron, keeping distance between them constant.